Searching for 4$\alpha$ linear-chain structure in excited states of $^{16}$O with a covariant density functional theory

A study of 4$\alpha$ linear-chain structure (LCS) in high-lying collective excitation states of $^{16}$O with a covariant density functional theory is presented. The low-spin states are obtained by configuration mixing of particle-number and angular-momentum projected quadrupole deformed mean-field states with generator coordinate method (GCM). The high-spin states are determined by cranking calculations. These two calculations are based on the same energy density functional PC-PK1. We have found the LCS candidate in both high-lying low-spin GCM states and cranking high-spin states with similar moment of inertia and band-head energy, which are estimated to be around 0.11 MeV and 30 MeV, respectively. The intrinsic configuration is considered to be the one that 4$\alpha$ clusters stay along a common axis and nucleons occupy the $(s)^4(p)^4(d)^4(f)^4$ configurations in a nonlocal way. Moreover, our results indicate that the spin and orbital angular momenta of all nucleons are parallel in the LCS states but the sum of spin-orbit splitting energies turns out to be much smaller than that of shell-like state. Besides, our fully microscopic GCM calculation has reproduced the excitation energies and $B(E2)$ values rather well for the rotational band built on the second $0^+$ state which has been previously considered to have $^{12}$C+$\alpha$ structure, and the dominant configuration turns out to be four $\alpha$ clusters with "kite"-like shape.